Abstract

The $q$-semicircular distribution is a probability law that
interpolates between the Gaussian law and the semicircular law. There
is a combinatorial interpretation of its moments in terms of matchings
where $q$ follows the number of crossings, whereas for the free
cumulants one has to restrict the enumeration to connected matchings.
The purpose of this article is to describe combinatorial properties of
the classical cumulants. We show that like the free cumulants, they
are obtained by an enumeration of connected matchings, the weight
being now an evaluation of the Tutte polynomial of a so-called
crossing graph. The case $q=0$ of these cumulants was studied by
Lassalle using symmetric functions and hypergeometric series. We show
that the underlying combinatorics is explained through the theory of
heaps, which is Viennot's geometric interpretation of the
Cartier-Foata monoid. This method also gives a general formula for
the cumulants in terms of free cumulants.